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A while ago this interesting question was asked Derived Algebraic Geometry and Chow Rings/Chow Motives.

Primary question: Have there been any recent developments/advances on the above question? If not, what are some of the main obstacles in preventing one from defining Chow Motives in a derived setting? References around this area would be wonderful. From what I understand a major hurdle is that a "nice" notion of a sub-scheme has not been defined in derived algebraic geometry.

General question: Elaboration: Has anyone worked on a different kind of "theory of motives"? In a similar vein to the development of prismatic cohomology which "unifies" p-adic cohomology theories in the sense that one can somewhat recover other p-adic cohomology theories from prismatic cohomology or Deligne's mixed Hodge structures in the complex case. These are definitely not as nice as motives since they are not universal. Question: The language of infinity categories is crucial in the formulation of prismatic cohomology. Has anyone attempted to unify prismatic cohomology with mixed Hodge structures using higher categorical methods or employed its methods to develop a cohomology that "recovers" any other cohomology? If not, what are the obstacles?

I hope this makes more sense.

I hope my set of questions makes sense. Thank you so much.

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    $\begingroup$ It is still unknown if Chow groups can be defined using derived geometry, but I don't think it is expected: the natural Chow-like theory that arises from derived cycles is algebraic cobordism, not Chow groups. I don't really understand the context of the second question: many categories of motives that have been defined (e.g. Voevodsky's) are $\infty$-categories, so naturally the latter are ubiquitous in papers about motives. $\endgroup$
    – Marc Hoyois
    Commented Oct 3 at 7:12
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    $\begingroup$ @MarcHoyois I have seen one lecturer propose a combination of cobordism and Lawson homology as an ersatz cycle theory (you helped me understand this proposal some time ago). $\endgroup$
    – Jason Starr
    Commented Oct 3 at 12:43
  • $\begingroup$ I hope the above is clearer since first was confusing. Also, I hope I'm understanding the theory developed by Bhatt and others correctly. :) $\endgroup$
    – Luqman Waheeduddin
    Commented Oct 3 at 21:27

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